Are neurons instantly feed forward when input arrives?
It is unclear what kind of network your are referring to, there is not a single neural-network model so conceivable both cases could exist and serve some purpose, yet if you are looking for one that emulates nature and real neurons, then you are missing at least 2 ingredients ( time and the mechanisms of resting potentials and refractory periods), which in turn introduce new computations to the neural network.
This is your network graph if I got it right:

The calculation in a neural network without refractory periods and resting potentials without time, would instantaneously modify the weight of your node4 layer 5 (n4-L5) if there is input in column 3:

Additional inputs on other columns would just add up unless you have some other explicit computation on any layer.
If you wanted to emulate a Neuron, each node would need to have a resting potential, that is: a level above which it will fire ( in the above example zero), and a refractory period:  a time before it would fire again, as well as a clock to keep it in sync, this would be a crude realtime fascimile:
http://codepen.io/k3no/pen/WRQbYV

A common alternative is to use sequential phases or steps.
Reference/source
Artificial Intelligence: A Modern Approach, by S. Russell and P. Norvig. Deals with a general step approach to A.I.
Gateway to Memory, by Mark A. Gluck and Catherine E. Myers Presents a great and readable introduction to modeling neural networks.
I published the little neural network model in a medium article : Memory and the machine, relevant sources are there.